The key to this construction (as well as the one in [1]}) is the existence of expanding solitons. For
trivalent vertices, their existence and uniqueness was proved by Schnürer and Schulze [2]}. For
more general vertices, their existence and classification was obtained by the second and fourth authors
here [3]}. One striking feature and outcome of this is that if any interior vertex has valence greater
than 3, then the flow is ill-posed: there exists more than one solution emanating from the given
initial configuration. The chosen soliton at each \(p\)  emerges in a very natural way, as we shall explain.
This illustrates yet again that nonuniqueness is endemic in the realm of flows on spaces with singular
initial data.
